Qualitative possibility theory
Qualitative possibility theory presupposes an ordinal representation of uncertainty in the form of a plausibility relation between possible worlds, that can be projected on a totally ordered qualitative scale. Possibilistic logic is then an extension of classical logic to knowledge bases where formulas are ordered according to their levels of certainty modelled by necessity measures. On top of possibility and necessity degrees induced by the plausibility ordering, two other set-functions can be defined, representing actual or guaranteed possibility and potential necessity.
- Belief revision : The AGM approach to theory revision is central in Artificial Intelligence and it implies a representation of priorities between formulas in agreement with possibility theory.
- Spohn’s rank functions encode possibility measures, under a specific format (by means of natural integers). This approach comes down to a system of infinitesimal probabilities.
- Non-monotonic reasoning : the inconsistency-tolerant inference relation in possibilistic logic is actually characterised by the properties of non-monotonic inference relations after Gabbay, Lehmann and Makinson.
More recently formal links between qualitative possibility theory and other topics in information processing and knowledge representation have been laid bare:
- Links with Formal Concept Analysis : The four set functions at work in possibility theory have counterparts in terms of Galois-like connections instrumental in FCA, which leads to new developments of FCA.
- Links with modal logic : The all-or-nothing version of possibility theory is exactly captured by a fragment of the modal logic KD called MEL, whose semantics in in terms of simple epistemic states. On this basis, a conjoint generalisation of possibilistic logic and MEL called generalized possibilistic logic has been devised. It yields an expressive multimodal logic that can encode answer-set programming and the equilibrium logic of Pierce, and logics of all I know. It enables to clarify the semantic content of formulas in such languages.
- Links with three-valued logics of incomplete information (noticeably Kleene logic) where the third truth-value refers to the idea of unknown, and some paraconsistent three-valued logics (noticeably Priest logic) where the third truth-value refers to the idea of contradiction. We could show that all such logics are captured by the MEL logic.
- Links with Belnap logic of contradiction: in fact it can be captured by monotonic all-or-nothing set functions and generalised by means of qualitative fuzzy measures measuring support in favor and against propositions.
- Links with the square of oppositions, a fundamental pattern of Aristotle logic.
In the long range, we aim at building a qualitative framework for uncertainty and qualitative information fusion, general enough to articulate qualitative extensions of possibility theory (qualitative capacities), some classes of non-regular modal logics, and some inconsistency-tolerant logics such as Belnap logic.
- Zina Ait-Yakoub, Yassine Djouadi, Didier Dubois, Henri Prade. Asymmetric Composition of Possibilistic Operators in Formal Concept Analysis: Application to the Extraction of Attribute Implications from Incomplete Contexts. International Journal of Intelligent Systems, Wiley, Vol. 32 N. 12, p. 1285-1311, 2017.
- Mohua Banerjee, Didier Dubois. A simple logic for reasoning about incomplete knowledge. In : International Journal of Approximate Reasoning, Elsevier, Vol. 55, pp. 639-653, 2014. (pdf)
- Mohua Banerjee, Didier Dubois, Lluis Godo, Henri Prade. On the relation between possibilistic logic and modal logics of belief and knowledge. Journal of Applied Non-Classical Logics 27(3-4): 206-224 (2017)
- Salem Benferhat, Didier Dubois, Henri Prade. Possibilistic and standard probabilistic semantics of conditional knowledge bases. In : Journal of Logic and Computation, Oxford University Press, Vol. 9 N. 6, pp. 873-895, 1999.
- Salem Benferhat, Didier Dubois, Henri Prade, Mary-Anne Williams. A framework for iterated belief revision using possibilistic counterparts to Jeffrey’s rule. In : Fundamenta Informaticae, IOS Press, Special issue Methodologies for intelligent systems, Vol. 99 N. 2, pp. 147-168, 2010.
- Claudette Cayrol, Didier Dubois, Fayçal Touazi. Symbolic possibilistic logic: completeness and inference methods. Dans / In : Journal of Logic and Computation, Oxford University Press, Vol. 28 N. 1, p. 219-244, 2018.
- Claudette Cayrol, Didier Dubois, Fayçal Touazi. Possibilistic reasoning from partially ordered belief bases with the sure thing principle. Dans / In : Journal of Applied Logic – IfCoLoG Journal of Logics and their Applications, College Publications, UK, Vol. 5 N. 1, p. 5-40, 2018.
- Davide Ciucci, Didier Dubois. A modal theorem-preserving translation of a class of three-valued logics of incomplete information. In : Journal of Applied Non-Classical Logics, Taylor & Francis Group, Vol. 23 N. 4, pp. 321-352, 2013. (pdf)
- Davide Ciucci, Didier Dubois. A capacity-based framework encompassing Belnap-Dunn logic for reasoning about multisource information. Int. J. Approx. Reasoning 106: 107-127 (2019)
- Didier Dubois. Belief structures, possibility theory and decomposable confidence measures on finite sets. In : Computers and Artificial Intelligence, Institute of Informatics, SAS, Bratislava – Slovaki, Vol. 5 N. 5, pp. 403-416, 1986.
- Didier Dubois. On Ignorance and Contradiction Considered as Truth-Values. In : Logic Journal of the IGPL, Oxford University Press, Vol. 16 N. 2, pp. 195-216, 2008.
- Didier Dubois. Reasoning about ignorance and contradiction: many-valued logics versus epistemic logic. In : Soft Computing, Springer-Verlag, Vol. 16 N. 11, pp. 1817-1831, 2012.
- Didier Dubois, Hélène Fargier, Henri Prade. Ordinal and Probabilistic Representations of Acceptance. In : Journal of Artificial Intelligence Research, AAAI Press, Vol. 22, pp. 23-56, 2004. (pdf)
- Didier Dubois, Jérôme Lang, Henri Prade. Possibilistic logic. In : Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 3. D.M. Gabbay, C.J. Hogger, J.A Robinson, D. Nute (Eds.), Oxford University Press, pp. 439-513, 1994.
- Didier Dubois, Emiliano Lorini, Henri Prade. The Strength of Desires: a Logical Approach. Dans / In : Minds and Machines, Springer, Vol. 27 N. 1, p. 199-231, 2017.
- Didier Dubois, Henri Prade. Possibility theory and formal concept analysis: Characterizing independent sub-contexts. In : Fuzzy Sets and Systems, Elsevier, Vol. 196, pp. 4-16, 2012. (pdf)
- Didier Dubois, Henri Prade, Agnés Rico. Representing qualitative capacities as families of possibility measures. Dans / In : International Journal of Approximate Reasoning, Elsevier, Vol. 58, p. 3-24, 2015. (pdf)
- Didier Dubois, Henri Prade, Agnés Rico. Graded cubes of opposition and possibility theory with fuzzy events. Dans / In : International Journal of Approximate Reasoning, Elsevier, Vol. 84, p. 168-185, 2017.
- Didier Dubois, Henri Prade, Steven Schockaert. Generalized possibilistic logic: Foundations and applications to qualitative reasoning about uncertainty. Dans / In : Artificial Intelligence, Elsevier, Vol. 252, p. 139-174, 2017.
- Didier Dubois, Giovanni Fusco, Henri Prade, Andrea Tettamanzi. Uncertain logical gates in possibilistic networks: Theory and application to human geography. Dans / In : International Journal of Approximate Reasoning, Elsevier, Vol. 82, p. 101-118, 2017.
- Fayçal Touazi, Claudette Cayrol, Didier Dubois Possibilistic reasoning with partially ordered beliefs.Journal of Applied Logic, Elsevier, Vol. 13 N. 4, p. 770-798, 2015. (pdf)